In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various formulas and is also given by the alternating harmonic series. The decimal value of the natural logarithm of 2 truncated at 30 decimal places is given by:

The logarithm of 2 in other bases is obtained with the formula

The common logarithm in particular is ()

The inverse of this number is the binary logarithm of 10: ().

By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number. It is also contained in the ring of algebraic periods.

Series representations

Rising alternate factorial This is the well-known "alternating harmonic series".

Binary rising constant factorial

Other series representations

using (sums of the reciprocals of decagonal numbers)

Involving the Riemann Zeta function

( is the Euler–Mascheroni constant and Riemann's zeta function.)

BBP-type representations

(See more about Bailey–Borwein–Plouffe (BBP)-type representations.)

Applying the three general series for natural logarithm to 2 directly gives:

Applying them to gives:

Applying them to gives:

Applying them to gives:

Representation as integrals The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:

Other representations The Pierce expansion is

The Engel expansion is

The cotangent expansion is

The simple continued fraction expansion is , which yields rational approximations, the first few of which are 0, 1, 2/3, 7/10, 9/13 and 61/88.

This generalized continued fraction: , also expressible as

The following continued fraction representation (J.L.Lagrange) gives (asymptotically) 1.53 new correct decimal places per cycle: A. N. Khovanski, The applications of continued fractions and their Generalisation to problemes in approximation theory,1963, Noordhoff, Groningen, The NetherlandsMathematical Analysis Functions, Limits, Series, Continued Fractions. Edited by L.A. Lyusternik and A. R. Yanpol’Skii Translated by D. E. Brown, Translation edited by E. Spence, 1965, p.273. Pergamon Press, Oxford, London, Edinburgh, New York

or

Bootstrapping other logarithms Given a value of , a scheme of computing the logarithms of other integers is to tabulate the logarithms of the prime numbers and in the next layer the logarithms of the composite numbers based on their factorizations

This employs PrimeApproximate natural logarithmOEIS 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97

In a third layer, the logarithms of rational numbers are computed with , and logarithms of roots via .

The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers close to powers of other numbers is comparatively easy, and series representations of are found by coupling 2 to with logarithmic conversions.

Example If with some small , then and therefore

Selecting represents by and a series of a parameter that one wishes to keep small for quick convergence. Taking , for example, generates

This is actually the third line in the following table of expansions of this type:

1 3 1 2 = … 1 3 2 2 − = −… 2 3 3 2 = … 5 3 8 2 − = −… 12 3 19 2 = … 1 5 2 2 = … 3 5 7 2 − = −… 1 7 2 2 = … 1 7 3 2 − = −… 5 7 14 2 = … 1 11 3 2 = … 2 11 7 2 − = −… 11 11 38 2 = … 1 13 3 2 = … 1 13 4 2 − = −… 3 13 11 2 = … 7 13 26 2 − = −… 10 13 37 2 = … 1 17 4 2 = … 1 19 4 2 = … 4 19 17 2 − = −… 1 23 4 2 = … 1 23 5 2 − = −… 2 23 9 2 = … 1 29 4 2 = … 1 29 5 2 − = −… 7 29 34 2 = … 1 31 5 2 − = −… 1 37 5 2 = … 4 37 21 2 − = −… 5 37 26 2 = … 1 41 5 2 = … 2 41 11 2 − = −… 3 41 16 2 = … 1 43 5 2 = … 2 43 11 2 − = −… 5 43 27 2 = … 7 43 38 2 − = −…

Starting from the natural logarithm of one might use these parameters: 10 2 3 10 = … 21 3 10 10 = … 3 5 2 10 = … 10 5 7 10 − = −… 6 7 5 10 = … 13 7 11 10 − = −… 1 11 1 10 = … 1 13 1 10 = … 8 13 9 10 − = −… 9 13 10 10 = … 1 17 1 10 = … 4 17 5 10 − = −… 9 17 11 10 = … 3 19 4 10 − = −… 4 19 5 10 = … 7 19 9 10 − = −… 2 23 3 10 − = −… 3 23 4 10 = … 2 29 3 10 − = −… 2 31 3 10 − = −…

Known digits This is a table of recent records in calculating digits of . As of December 2018, it has been calculated to more digits than any other natural logarithm of a natural number, except that of 1.

Date Name Number of digits January 7, 2009 A.Yee & R.Chan 15,500,000,000 February 4, 2009 A.Yee & R.Chan 31,026,000,000 February 21, 2011 Alexander Yee 50,000,000,050 May 14, 2011 Shigeru Kondo 100,000,000,000 February 28, 2014 Shigeru Kondo 200,000,000,050 July 12, 2015 Ron Watkins 250,000,000,000 January 30, 2016 Ron Watkins 350,000,000,000 April 18, 2016 Ron Watkins 500,000,000,000 December 10, 2018 Michael Kwok 600,000,000,000 April 26, 2019 Jacob Riffee 1,000,000,000,000 August 19, 2020 Seungmin Kim 1,200,000,000,100 September 9, 2021 William Echols 1,500,000,000,000 February 12, 2024 Jordan Ranous 3,000,000,000,000 November 15, 2025 Mamdouh Barakat 3,100,000,000,000

See also Rule of 72#Continuous compounding, in which figures prominently Half-life#Formulas for half-life in exponential decay, in which figures prominently Erdős–Moser equation: all solutions must come from a convergent of .

References

External links